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Given a super-critical branching random walk on $\mathbb{R}$ started from theorigin, let $Z\_n(\cdot)$ be the counting measure which counts the number ofindividuals at the $n$-th generation located in a given set. Under some mildconditions, it is known in \cite{B90} that for any interval $A\subset\mathbb{R}$, $\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}$ converges a.s. to$\nu(A)$, where $\nu$ is the standard Gaussian measure. In this work, weinvestigate the convergence rates of$$\mathbb{P}\left(\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}-\nu(A)>\Delta\right),$$for $\Delta\in (0, 1-\nu(A))$, in both Schr{\"o}der case and B{\"o}ttcher case.

On large deviation probabilities for empirical distribution of branching random walks: Schr{\"o}der case and B{\"o}ttcher case